# Appendix: Syntax summary¶

The following code illustrates the syntax of the various blocks used in IDP-Z3.

T denotes a type, c a constructor, p a proposition or predicate, f a constant or function. The equivalent ASCII-only encoding is shown on the right.

```vocabulary V {
type T
type T ≜ {c1, c2, c3}                     type T := {c1, c2, c3}
type T ≜ constructed from {c1, c2(T1, f:T2)}
type T ≜ {1,2,3}                          type T := {1,2,3}
type T ≜ {1..3}                           type T := {1..3}
// built-in types: 𝔹, ℤ, ℝ, Date, Concept Bool, Int, Real, Date, Concept

p : () → 𝔹                                p: () -> Bool
p1, p2 : T1 ⨯ T2 → 𝔹                      p1, p2: T1*T2 -> Bool
f: T → T                                  f: T -> T
f1, f2: Concept[T1->T2] → T               f1, f2: Concept[T1->T2] -> T

[this is the intended meaning of p]
p : () → 𝔹

var x ∈ T                                 var x in T
import W
}

theory T:V {
(¬p1()∧p2() ∨ p3() ⇒ p4() ⇔ p5()) ⇐ p6(). (~p1()&p2() | p3() => p4() <=> p5()) <= p6().
p(f1(f2())).
f1() < f2() ≤ f3() = f4() ≥ f5() > f6().  f1() < f2() =< f3() = f4() >= f5() > f6().
f() ≠ c.                                  f() ~= c.
∀x,y ∈ T: p(x,y).                         !x,y in T: p(x,y).
∀x ∈ p, (y,z) ∈ q: q(x,x) ∨ p(y) ∨ p(z).  !x in p, (y,z) in q: q(x,x) | p(y) | p(z).
∃x ∈ Concept[()→B]: \$(x)().               ?x in Concept[()->B]: \$(x)().
∃x: p(x).                                 ?x: p(x).

f() in {1,2,3}.
f() = #{x∈T: p(x)}.                       f() = #{x in T: p(x)}.
f() = min{  f(x) | x in T: p(x) }.
f() = sum{{ f(x) | x in T: p(x) }}.
if p1() then p2() else p3().
f1() = if p() then f2() else f3().

p ≜ {1,2,3}.                             p := {1,2,3}.
p(#2020-01-01) is enumerated.
p(#TODAY) is not enumerated.

{ p(1). }
{ (co-induction)
∀x∈T: p1(x) ← p2(x).                    !x in T: p1(x) <- p2(x).
f(1)=1.
∀x: f(x)=1 ← p(x).                      !x: f(x)=1 <- p(x).
∀x: f(x)≜1 ← p(x).                      !x: f(x):=1 <- p(x).
}

[this is the intended meaning of the rule]
p().
}

structure S:V {
p ≜ false.                               p := false.
p ≜ {1,2,3}.                             p := {1,2,3}.
p ≜ {0..9, 100}.                         p := {0..9, 100}.
p ≜ {#2021-01-01}.                       p := {#2021-01-01}.
p ≜ {(1,2), (3,4)}.                      p := {(1,2), (3,4)}.
p ≜ {                                    p := {
1 2                                       1 2
3 4                                       3 4
}.                                        }.

f ≜ 1.                                   f := 1.
f ≜ {→1} .                               f := {-> 1}.
f ≜ {1→1, 2→2}.                          f := {1->1, 2->2}.
f ≜ {(1,2)→3} else 2.                    f := {(1,2)->3} else 2.
f ⊇ {(1,2)→3}.                           f >> {(1,2)->3}.
}

display {
goal_symbol ≜ {`p1, `p2}.                goal_symbol := {`p1, `p2}.
hide(`p).
expand ≜ {`p}.                           expand := {`p}.
view() = expanded.
optionalPropagation().
}

procedure main() {
pretty_print(model_check    (T,S))
pretty_print(model_expand   (T,S))
pretty_print(model_propagate(T,S))
pretty_print(minimize(T,S, term="cost()"))
}
```

It is possible to use English connectives to create expressions:

```for all T x:                                 ∀ x ∈ T:
there is a T x:                              ∃ x ∈ T:

p() or q()                                   p() ∨ q()
p() and q()                                  p() ∧ q()
if p(), then q()                             p() ⇒ q()
p() are sufficient conditions for q()        p() ⇒ q()
p() are necessary conditions for q()         p() ⇐ q()
p() are necessary and sufficient conditions for q() p() ⇔ q()
p() is the same as q()                       p() ⇔ q()

x is y                                       x = y
x is not y                                   x ≠ y
x is strictly less than y                    x < y
x is less than y                             x ≤ y
x is greater than y                          x ≥ y
x is strictly greater than y                 x > y

the sum of f(x) for each T x such that p(x)  sum{{f(x) | x∈T: p(x) }}
p() if q().                                  p() ← q().
```